Theorem sucexeloni 7787

Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7788 does not require ax-un 7713. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)

Assertion

Ref	Expression
sucexeloni	⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloni

Step	Hyp	Ref	Expression
1		eloni 6351	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsuci 7786	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		elex 3474	. 2 ⊢ (suc 𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
5		elong 6349	. . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
6	5	biimparc 483	. 2 ⊢ ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On)
7	3, 4, 6	syl2an 605	1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  Vcvv 3453  Ord word 6340  Oncon0 6341  suc csuc 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6344  df-on 6345  df-suc 6347

This theorem is referenced by:  onsuc  7788  1on  8444  2on  8445